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Beyond the Efron–Buchta Identities: Distributional Results for Poisson Polytopes

Discrete & Computational Geometry volume 53, pages 226–244 (2015)Cite this article

Abstract

Let \(\varPi \) be a random polytope defined as the convex hull of the points of a Poisson point process. Identities involving the moment-generating function of the measure of \(\varPi \), the number of vertices of \(\varPi \), and the number of non-vertices of \(\varPi \) are proven. Equivalently, identities for higher moments of the mentioned random variables are given. This generalizes analogous identities for functionals of convex hulls of i.i.d points by Efron and Buchta.

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Acknowledgments

M. Beermann was supported in part by the FWF project P 22388-N13, ‘Minkowski valuations and geometric inequalities’. We are grateful to an anonymous referee for careful reading of the manuscript and numerous helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Osnabrueck, Albrechtstr. 28a, 49076, Osnabrueck, Germany

    Mareen Beermann & Matthias Reitzner

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  1. Mareen Beermann
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  2. Matthias Reitzner
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Correspondence to Matthias Reitzner.

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Beermann, M., Reitzner, M. Beyond the Efron–Buchta Identities: Distributional Results for Poisson Polytopes. Discrete Comput Geom 53, 226–244 (2015). https://doi.org/10.1007/s00454-014-9649-7

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  • DOI: https://doi.org/10.1007/s00454-014-9649-7

Keywords

  • Poisson polytope
  • Random polytope
  • Generating function
  • Efron’s identity

Mathematics Subject Classification

  • Primary 60D05
  • Secondary 60G55
  • 52A22